Métivier groups, introduced by Métivier [1]} in his study
of analytic hypoellipticity, are two-step nilpotent Lie groups satisfying a
non-degeneracy condition. They are also characterized by the property that
the quotients with respect to the hyperplanes contained in the centre are
general Heisenberg groups. H-type groups, introduced by Kaplan [2]}, are typical examples of Métivier groups, but there are many Métivier groups which are not isomorphic to H-type groups, for example, see
[3]}. Heisenberg groups are the simplest Métivier
groups. Casarino and Ciatti [4]} investigated the spectral resolution
of the sub-Laplacian on Métivier groups, and proved a Stein-Tomas
restriction theorem in terms of the mixed norms. So, we can use the the
spectral decomposition of the sub-Laplacian to define the Riesz means and
the bilinear Riesz means on Métivier groups. Mauceri [5]} and Müller [6]} obtained the same results on the \(L^{p}\) -boundedness
of the Riesz means on Heisenberg groups by using different methods. We
proved the \(L^{p_{1}}\times L^{p_{2}}\rightarrow L^{p}\)  boundedness of the
bilinear Riesz means on Heisenberg groups [7]}. To extend these
results to general Métivier groups, we notice that there are two
essential difficulties: one is that Heisenberg groups have one dimensional
centre but the centre dimension of Métivier groups is in general bigger
than one; the other is that on H-type groups, the non-degeneracy condition
becomes a better orthogonality condition, which is not true on general Métivier groups. Therefore, in this paper, we shall introduce some different
techniques from those used in [5]} or [6]} and [7]} to
obtain the \(L^{p}\) - boundedness of the Riesz means and \(L^{p_{1}}\times L^{p_{2}}\rightarrow L^{p}\)  boundedness of the bilinear Riesz means on Métivier groups.
